A symmetric quadratic ratio bounds cyclic ratio differences

Proved

WorkbookSource.base_13018

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

For any $a,b,c >0$ , prove that $\frac{a^2+b^2+c^2}{ab+bc+ca} \ge 1+ \frac{1}{3} \cdot \sum_{cyc} \left(\frac{a}{a+b}-\frac{b}{b+c}\right)^2$

Source: InternLM Lean-Workbook, record lean_workbook_13018 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13018 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) / (a * b + b * c + c * a) ≥ 1 + 1 / 3 * ( (a / (a + b) - b / (b + c))^2 + (b / (b + c) - c / (c + a))^2 + (c / (c + a) - a / (a + b))^2 )  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13018; Apache-2.0

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